Inverse Problems for mathematical models of quasistationary electromagnetic waves in anisotropic nonmetallic media with dispersion Текст научной статьи по специальности «Математика»

MSC 35R30, 35Q60, 35Q35

DOI: 10.14529/ mmp 180105

INVERSE PROBLEMS FOR MATHEMATICAL MODELS OF QUASISTATIONARY ELECTROMAGNETIC WAVES IN ANISOTROPIC NONMETALLIC MEDIA WITH DISPERSION

S.G. Pyatkov1'2, S.N. Shergin1

1Ugra State University, Khanty-Mansyisk, Russian Federation 2South Ural State University, Chelyabinsk, Russian Federation E-mail: [email protected], [email protected]

We consider inverse problems of evolution type for mathematical models of quasistationary electromagnetic waves. It is assumed in the model that the wave length is small as compared with space inhomogeneities. In this case the electric and magnetic potential satisfy elliptic equations of second order in the space variables comprising integral summands of convolution type in time. After differentiation with respect to time the equation is reduced to a composite type equation with an integral summand. The boundary conditions are supplemented with the overdetermination conditions which are a collection of functionals of a solution (integrals of a solution with weight, the values of a solution at separate points, etc.). The unknowns are a solution to the equation and unknown coefficients in the integral operator. Global (in time) existence and uniqueness theorems of this problem and stability estimates are established.

Keywords: Sobolev-type equation; equation with memory; elliptic equation; inverse problem; boundary value problem.

Introduction

We consider the problems arising in the description of propagation of both electromagnetic waves in anisotropic media [1] and nonstationary interior waves in an incompressible stratified rotating fluid [2]. The peculiarities of propagation of electromagnetic waves in anisotropic media are defined by the corresponding material equations. If the length of a wave is small as compared with space inhomogeneities that these equations can be written in the form accounting for time dispersion only and introducing the potentials of electric and magnetic fields E = -Vp(x,t) and H = -V^(x,t) and making some transformations we arrive at the equations (see [1], p. 28)

3 3

+ 4nKi *)wxi,xi = -4np + Fd, ^ (1 + 4nKi*)^XuXi = Fb, (1)

i=1 i=1

where Ki are the diagonal entries of the tensors of electric and magnetic susceptibilities and Ki*^(x, t) = /0 Ki(t — T)p(x, t) dr. Note that some model problems for nonstationary waves in media with anisotropic dispersion are reduced to integro-diflerential equations (1) with kernels of convolution operators of the form of a sine, a polynomial, or an exponential function. In these cases it is possible to reduce an initial vector systems of equations by introducing generalized potentials of quasistationary electric and magnetic fields to composite type equations (see [2]) of the form

3

Ps(dt)A$(x,t) + Pmi3 (dt)Y, Qxxj = F,

i,j=1

where Ps, Pmij are polynomials of degrees s and m, respectively. At the present article we examine inverse problems on recovering the coefficients ki for general equations of the form

m

Lou + Ki*LiU = f, (2)

i=1

where

n n

Lku aij(x,t)uxi,xj ai (x,t) + ak(x,t)u, (x,t) e Q = G x (0,T), G c Rn

ij=i i=i

The equation (2) is supplemented with the overdetermination conditions

V (u)(t) = ^ (t), (3)

where V are some functional (the conditions on them are described below), and the boundary conditions

Bu\s = g(x,t), S = dG x (0,T), (4)

where Bu = u or Bu = Y^n=1 Yi(x,t)uXi + a(x,t)u. Similar equations and systems of equations arise in elasticity (materials with memory) [3-5], physics (phase-field models, heat and mass transfer) [6,7], and in many other fields. The most known case is the case

Lo

case was studied in which L0 = dt — A or L0 = d\ — A, with A a generator of an analytic

Lo

in [13]. In the case of L0 = dt, we arrive at Gurtin-Pipkin-type models (see [14,15]). Probably, the elliptic case was not considered except for one model situation (see [16]), where n = 1. We establish global (in time) solvability of the problem (2) - (4) in Sobolev spaces.

1. Preliminaries

We employ the Sobolev spaces WpS(G) and Holder spaces Ca (G). The symbol Lp(0,T; H) (H is a Banach space) stands for the spaces of strongly measurable functions defined on [0, T] with values in H (see the definition of the function spaces, for instance, in [17]).

We assume below that r = dG e C2 (see the definition, for example [18, Sect. 1, Ch. 1]) and that the coefficients of the operators Lk (k = 0,1,... ,m) are real-valued and the operator L0 is elliptic, i. e., there exists a constant 50 > 0 such that

m

1°

i,j=l

> I2 v ^ e v (x,t) e Q.

We fix the parameter p > n (for simplicity) and suppose that

aij e C(Q), a°,a° e C([0,T]; LP(G)), aj e L(Q), a^aj e Lp(0,T; L^(G)), ak ,aj e L^(0,T; LP(G)), akot,ai e LP(Q) (ij = 1, 2,..,n, k = 1, 2,...,m), (5) Yi,Yit,*,*t e C1l(S) (i = l,...,n), I £I=11гПг1 > 5-1 > 0 v(x,t) e S,

Вестник ЮУрГУ. Серия «Математическое моделирование и программирование» (Вестник ЮУрГУ ММП). 2018. Т. 11, № 1. С. 44-59

where n = (n1, n2,..., nn) is the outward unit normal to S and ^ is a constant. The operator L0 is assumed invertible, i. e., the following theorem is valid.

Theorem 1. Let the condition (5) hold. The problem with a parameter

Lo(t)u = f (x), B(t)u\r = g(x), (6)

for every f E Lp(G) (p > n) and g E WpS0 (r) (so = 2 — 1/p in the ease of the Diriehlet conditions and s0 = 1 — 1/p the case of the oblique derivative problem) has a unique solution u E W2(G) satisfying the estimate

IMIw?(g) < c(\\f \\Lp(G) + \\g\\w;°(d),

where the constant c is independent of f,g,t E [0,T],

The claim of the theorem holds whenever ker L0 = 0. In particular, it suffices to require in the case of the Diriehlet boundary conditions that a0 < 0 a.e. in Q (see the maximum principle [19, Ch. 8]) and in the case of the oblique derivative problem that a0 < 0 a.e. in Q and a0 < 0 in ^^^e neighborhood about S (see Proposition 2.3.2 and Theorem 2.3.5 in [20]).

Corollary 1. As a direct corollary of the claim of the theorem, we have that if f (x,t) E C ([a,p ]; Lp (G)) an d g(x,t) E C ([a,p]; Ws0 (r)) (0 < < T) then the problem

L0(t)u = f (x,t), Bu\r = g(x,t), (7)

has a unique solution u E C([a,fi]; W2(G)) satisfying the estimate

\u\c (\a,0\;W2(G)) < c(\f \o([a,^];Lp (G)) + \g\c([a/i];W;° (T))) >

where the constant c is independent of f,g, a, ft. It is not also difficult to demonstrate that if there exist the generalized derivatives ft E Lp(a, ft; Lp(G)), gt E Lp(a, ft; Wp° (r)) then a solution to the problem (7) is differentiable with respect to t, ut E Lp(a, ft; W£(G)) and

L0(t)ut = ft(x, t) — L0tu, But\r = gt(x, t) — Btu\r, (8)

t E ( a, ft ) ( L0t , Bt

t L0, B)

< c(\\f Ile ([a,fi];Lp (G)) + \\g\\c([a,fi];W;° (r)) + Wft\\lp (a,fi;Lp (G)) + (r}})>

\Lp(a,l3;W 2(G)) + \M\c([a,l3]; W2(G)) <

(r)) + \\ft\\Lp (a,l; Lp (G)) . „„.„^^.„p

where the constant c is independent of a, ft, f, g.

Note that the conditions (5) imply that the coefficients ak ,ak ,ajk belong to the space C([0,T]; Lp(G)) after a possible modification on a set of zero measure. In what follows we assume this condition to be fulfilled.

Lemma 1. The following inequalities hold:

\\u * v\\Lp(0,y) < WuWLp(0,7)\M\li(0,7), \\u * v\\Lp(o,y) < Y1-1/pWuWLp(0,7)\M\Lp(0,y), (9)

. Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming

& Computer Software (Bulletin SUSU MMCS), 2018, vol. 11, no. 1, pp. 44-59

Mt)\\L^0r() < Y1-1'v\\ut\\Lp(0r(), u(0) = 0, (10)

IUWIImqy) < 7\\Ut\\Lp(0Y), u(0) = 0. (11)

The inequalities (9) are known (see, for instance, Lemma 3.1 in [6]). The inequalities (10), (11) are an obvious consequence of the Newton-Leibnitz formula.

Assume that a solution to the problem (2) - (4) possesses the property ut E Lp(0, T; Wp2(G)) and u E C([0,T]; W2(G)), the conditions (5) hold, g E C([0,T]; W.s(r)), and gt E Lp(0,T; W^0(r)), ki(t) E Lp(0,T) for all z, and f,ft E Lp(Q). Taking t = 0 in (2), we infer LQ(x, 0)u(x, 0) = f (x, 0). Applying Theorem 1, we can find the function u(x, 0) = uQ(x) as a solution to the problem (6) at t = 0. The boundary condition (3) yields

Buo(x)\r = g(x, 0). (12)

The condition (4) implies the necessary solvability condition

Vj(uo(x)) = ^(0) j = 1, 2,... ,m. (13)

t

„ t m m

(Lou)t + / y^ki(r )(Lut(x,t - r ) + Litu(t - r )) dr = ft - V] k()Li (x, 0)uq(x). (14) Jq i=i i=i

Construct an auxiliary function $(x,t) E C([0,T]; Wp(G)) such that $t E Lp(0, T; W^(G)), B= g(x, t), and $(x, 0) = uQ(x). Let $ be a solution to the problem (7). Making the change of variables u = v + we obtain the problem

t m

(Lov)t + /£ ki(r)(Liv)t(x,t - r) dr =

Q i=1

m t m

= - £ ki(t)Li(x, 0)uq(x) -J £ ki(r )(Li$)t(x, t - r) dr = fo,

i=1 Q i=1

(15)

Bv\r = 0, v(x, 0) = 0, (16)

Vj(v) = ф - Vj(Ф) = ф, j = 1, 2,... ,m. (17)

Theorem 2. Assume that f,ft E Lp(Q), the conditions (5), (12), (13) hold, g E C([0,T]; W/0 (Г)), gt E Lp (0,T; W/0 (Г)) (p > n), and ф E W}(0,T) (j = 1, 2,...,m). Then the problem (15) - (17) of determining the functions v,k1,... ,km from the class vt E Lp (0,T; W2(G)), v E C ([0,T]; W2(G)), ki E Lp (0,T) (г = 1, 2,...,m) is equivalent to the problem (2) - (4) of determining the functions u,k1,...,km such that ut E Lp (0,T; W2(G)), u E C ([0, T ]; W2(G)), an d ki E Lp (0,T) (г = 1, 2,...,m).

Proof. Actually, the arguments presented before the theorem show that if u,k1,... ,km is a solution to the problem (2) - (4) from the above-pointed class then the functions v,k1,... ,km is a solution to the problem (15) - (17). So it suffices to verify the converse statement. Let v, k1,..., km is a solution to the problem (15) - (17). Put u = v + Ф. The equalities ft = (L^)t and (15) imply that

t m

dt(Lou) + dt$ £ ki(r)Liu(x, t - т) dr = ft, (18)

0 i=l

Integrating this equality from 0 to t, we derive that

t m

L0u + J ki(r)Liu(x, t — r) dr = f (t) + L0u(x, 0) — f (x, 0). 0 i=1

Since 0 = v(x, 0) = u(x, 0) — $(x, 0) = u(x, 0) — u0(x), u(x, 0) = u0(x), the definition of the function u0, yields Lu(x, 0) — f (x, 0) = 0, i. e., the equality (2) holds. The validity of the condition (4) is obvious.

□

Let 0 < a < 3 < T. Define the space H(a, 3) as the space of functions v(x,t) such that vt e Lp(a, 3; W2(G)), v e C([a,3]; W2(G)), Bv\r = 0. Endow it with the norm \\v\n(a,/3) = \\v\\o({a,i3];W2(g)) + \\vt\\Lp(a,i3;W2(G))- In what follows, a norm of a vector is the sum of the norms of its coordinates.

Lemma 2. Let the conditions (5) hold. Then

\\(Ljv)t\\Lp{a,i3;Lp(G)) < c\\v\\H(afi) Vv e C([a, 3]; W2(G)): vt e Lp(a,3; Wp2(G)), (19)

\\Ljtv\\Lp{a,i3;Lp{G)) < c\\v\\c{Mw*AG)Vv e C([a, 3]; WP2(G)): vt E Lp(a,3; W^(G)), (20) where 0 < a < 3 < T and the constant c is independent of a, 3,j-

Proof. The proof is more or less obvious. As an example, we establish (20). The expression Ljtv contains the summands ajktvXiXk1 ajktvXk, and aj0tv. We have

I

\\ajktv XiXk (G) \\vx iXk \L^(a,I;Lp(G)) < Cjk (a,3)\\v\\H(a,l3). (21)

Similarly, we infer

\\aMvxk\\Lp(aILp(G)) < \\ait\\Lp(aILp(G))\\vxk\\lx (a,3)\\v\\H(a,[i), (22)

\a0tv\Lp(aILp(G)) < C0(a,3)№(a!)- (23)

Here we employ the embedding Wp(G) C L^(G) (see [17]). We can take the sum of the constants cjk(0,T), cj(0,T), c0(0,T) over all indices as the constant in (20).

□

Lemma 3. Let the conditions (5) hold. Then the following inequalities hold:

\\fkj(r)(Ljw)t(x, t — r) dr\\Lp(0,r,Lp(G)) < ciY1-1/p\\k3\Lp(0Y)\w\H(0Y), Y<T, (24) 0

\\ J kj (r)(Ljw)t(x,t — r) dr\\Lp(l,l+r,Lp(G)) < clY1-1/P\\kj\Lp(l,l+1)\w\H(0Y), l + Y < T, (25)

r t-l

\\ / kj (r)(Ljw)t(x,t — r) dr\\Lp(l,l+r,Lp(G)) < c1Y1-1/P\\kj\Lp(0,1)\w\H(l,l+1), l+Y < T> (26) 0

valid for every every w e C([a,3]; Wp(G)) such that wt e Lp(a,3; Wp(G)). The constant

c1 Y, l, j

Proof. То prove (24), we first use the Minkowski inequality inserting the norm in Lp(G) under the integral sign. Next, we apply Lemmas 2 and 3. To establish (25), we use the change of variables. We have

pt r-r

\\ / kj (r)(Ljw)t(x,t-r) dr\\Lp(l,l+r,Lp(G)) = \\ kj (r1 + l)(Ljw)t(x,r-ri) dTl\\Lp(0,r,Lp(G)).

Jl J 0

Thus, we obtain the integral of the form (24) which is estimated similarly. The estimate of the last integral after the change of variables is reduced to estimating the espression

r

\\ / kj (r )(Lj w)t(x,r + l - T) dr hp(0,T,Lp (G)), 0

which is again of the same form as the left-hand side of (24).

□

Consider the auxiliary equation

t m

L0vt + L0tv + f £ ki(r)(Liv)t(x,t - т) dr = f'0. (27)

0 i=1

Let QY = G x (0,7). Fix T0 < T.

Theorem 3. Assume that f0 E Lp(QTo) (p > n), the conditions (5) ho Id, and ki E Lp(0,T0) (г = 1, 2,... ,m). Then there exsists a unique solution to the problem (16), (27), such that v E H(0,T0). There exists а соnstant c> 0 independent of f and T0 such that a solution to the problem (16), (27) satisfies the estimate

\М\са0То]; W2(G)) + \vt\Lp(0,To ; W2(G)) < c\\f0 WLp(QT0).

Proof. We reduce the problem to an integral equation. From (27) we have

t S, m t

v(x, t) + L-1// £ ki(r)(Liv)s(x, £ - T) drdC = L-1 f f0(x, т) dr = fi, (28) 0 0 i=1 0

where the operator L-1f takes a function f onto a solution to the problem (7) with g = 0. First, we justify a local solvability. We have the equation

v + S (v) = f1 (29)

Estimate \\S(v)W#(0,7) (7 < T0). Corollary 1 and Lemma 3 yield

\\S(v)b(0,7) < C71-1/P||v||H(o,Y)||k||Lp(0,Y) < c71-1/p||v||н(0,YMLp(0,n), (30)

where the constant c is independent of 7 and k = (k1,k2,..., km). Hence, for 7 < 70 with 7^-1/pc\k\Lp(0T) = 1/2, we obtain that ||S(v)||H(0)7) < ||v||H(0)7)/2 and the fixed point

[0 , 70 ]

solvable on every of the segments [0, h(0 + r0], wit h b(0 <T0, l = 1, 2,..., r0 < min(70 ,T0 -

l70 )

[0 , l70 ]

v(x,t) + S0(v) = f1 - S(v) + S0(v) = f2, (31)

0, t < Iyq,

sq(v) = < L-1 f ¡H-110 E ki(r)(Liv)t(x,£ - r) drd£, t E (lYQ,lYQ + tq) ' (32)

i=1

It is easy to make sure that the expression -S(v) + SQ(v) contains the values of the function v on the segment [0, k^o] only and thereby this expression is an already known function. By Corollary 1 and Lemma 3, the operator SQ(v) admits the estimate

\\ SQ (v)\H(Q,ljo+ro) < ,ljo+To) \\ k\Lp(Q,jo) < \v\h(q,11o+to) / 2 (33)

(as is easily seen, we can assume that the constant c in (33) and that in (30) coincide). Hence, the equation (31) is solvable. Obviously, a solution to the equation (31) with the right-hand side f2 is an extension of a solution to the equation (29) on the segment [iyq,iy0 + tq].

□

2. Main Results

Write out additional conditions on the data of the problem. We examine the problem (2) - (4) assuming that the functional Vj meet the conditions

Vj E L(Wp(G), R), Vj (uo) = 4j (0),j = 1, 2,... ,m. (34)

The symbol L(A,B) for given spaces A, B stands for the space of linear continuous operators defined on A with values in B.

Well-posedness conditions. Assume that B is the matrix with entries bij = Vi(L--1 Lj(x, 0)uQ(x)) and there exists a constant 62 > 0 such that

\ detB\ > 62 yt E [0,T], (35)

where L-1f is a solution vQ to the problem L0v0 = f, BvQ\r = 0.

f, ft E Lp(Q)

hold, g E C([0,T]; WS0(r)), gt E Lp(0,T; Wp0(r)) (p > n), and 4 E W}(0,T) (4 = (41,42,... ,4m)- Then there exists a unique solution to the problem (2) - (4) such that ut E Lp(0,T; Wf(G)), u E C([0,T]; W^(G)), k E Lp(0,T), k = (h,k2,...,km). For any two solutions (u1, k1), (u2, k2) to the problem (2) -(4) relating to the data fi,gi,4i (z = 1, 2) satisfying the conditions of the theorem, there is the estimate

\\u1 - u2\\c([Q,T];W2(G)) + \\u1t - u2t\\Lp(Q,T;W2 (G)) + YJZ1 \\k1 - k2\\Lp(Q,T) <

(36)

< c(\\f1 - Mw^QT; Lp(G)) + \\g1 - g2\\Lp(Q,T;W;° (r)) + \\41 - 42\w1(q,T)) , c

spaces and the constants in the condition (35). Proof. Consider the equivalent problem (see Theorem 2)

t m m

(Lov)t + /E ki(r)((Ljv)t(x,t-r) + (Lj$)t(x,t-r)) dr +£ kj(t)Lj(x, 0)uo(x) = 0, (37) o j=1 j=1

Bv\r = 0, v(x, 0) = 0, (38)

Vj(v) = fj, j = 1, 2,... ,m. (39)

Construct a system for determining the functions ki. Inverting the operator L0, we arrive at the equation

t m

vt(x, t) + L-lLQtV + L-1 f E kj (t)((Ljv)t(x, t - r) + (Lj$)t(x, t - r)) dr+

0 ^ m (40)

+vo(v) = - E kj(t)L- lLj(x, 0)uo(x), j=i

where L-1/ takes a function f onto a solution to the problem (7) with g = 0 and the function v0(v) is zero in the case of the Dirichlet boundary conditions while v0(v) is a solution to the problem L0v0 = 0, Bv0 = -Btv in the case of the oblique derivative

Bt t

B

t £ m

Lov(x,t) = -J J E k3 (т )(т )((Lj v)t(x,£ - т ) + (Lj Ф)^^ - т )) drd^-0 0 j=1

m t

-E fkj (t ) dÇLj (x, 0)uo(x) = G(v), j=1 0

(41)

v(x,t) = L-1G(v). (42)

Applying the functional to (40), we obtain

t m

Фи + Фi(L-1 Lotv) + VAL-1 f E kj(t)(Ljv)t(x,t - т) dT) + ФМ =

0 j=1

m t m

11

(43)

- E kj (t)Vi(L- Lj (x, 0)uo(x)) - ViL-1! E kj (t )(Lj Ф)^ - т ) dT ). j=1 0 j=1

In view of (35) this equality implies that

k = B-1F — B-1A(k), (44)

where F has the coordinates Fi = —'it E W^(0,T) (i = 1, 2,...,m) and the operator A(k) the coordinates

t

/■m

J2kj(r)((Ljv)t(x,t—T) + (Lj§)t(x,t—r)) dr^(vo(v)), o j=1

where v = v(k) is a solution to the problem (37), (38). Thus, we have the system for the vector-function k = (k1,k2, ..,km). Prove its solvability. Note that by Theorem 3 (used on the segment [0,7]), for a given k E Lp(0,y) (7 < T), we can uniquely determine the function v = v(k) E H(0,7) as a solution to the equation (42). Establish some

estimates. Let R = 2\\B 1F\\Lp(Q ,T). We look for a vector k in the ball BR = {k E Lp(0,y) ■ I k\\Lp(Q,Y) < R}. Let k E BR Estimate the quantity \\L-1G(v)\\H(0y). We have

t S

m

G(v) - G(0) = J J ^kj (r )(r )(Lj v)t(x, £ - r) drd£ 0 0 j=1

Corollary 1, Lemma 4, and the inequalities (11) imply that \\L-\G(v) - G(0))\\h(qy) = \\L-\G(v) - G(0))\\h(qy) < C1Y1-1/p\\khp(Qy)\\v\\h(qy), (45)

\l^1g(0)\h(Qy) < C1Y^II k\\Lp(oy)\\$\\hQy) + C2\\kh^) . (46)

Here c1, c2 are some constants independent of y and the unknowns. Thus, if

^c, R =1/2, (47)

then, for y < Yf0, a solution to the equation (42) satisfies the estimate

\\v\\h(Qy) < 2\\k\\Lp(Qy)(c1\$\h(Q,T)Yi~1/p + C2) = 2\Ik\\Lp(Qy)C3, (48)

which can also rewritten as

\\v\\h(q y) < 2Rc3. (49)

Proceed with the estimates for the summands in (44). Corollary 1 and the trace theorems [17, Sect. 4.7] yield

\\vQ\\Lp(a^,W2(G)) < c\\Btv\\Lp(a l^0 (r)) < c4v\\Lp(a ^W^p (r)) < c^\v\Lp(a ,l3;W2(G)), (50)

where the constant c5 is independent of 0 < a < 3 < T. Let kQ = maxi \\Vi\L(^2(G) R). In view of (10), (34), (48), (50), we infer

E™1 \\ Vi(vQ )|Lp(QY) < KQc5\\v\\Lp(Q,r,W2(G)) < c6Yl-l/P\\vt\\Lp(Q,r,W2(G)) <

(51)

< 2c3c&Yl-1/p\\k\\Lp(Qy) . Using the arguments those in the derivation of (20), (10), Corollary 1, and (48), we obtain

j \\Vj (L-lLQtv)\\Lp(Q, y) < KQc7\\v\\Lp(Q, y;W2(G)) < c8Y1-1/p\\vt\\Lp(Q, j;W2(G)) <

(52)

< 2csc&y 1-1/p\\k\\Lp(o,y),

where the constant c8 is independent of y and k = (k1,k2,... ,km). Next, using (34), Corollary 1, and Lemma 3, we have

t m

Em=1 \\Vi(L0-1 / E kj(r)((Ljv)t(x,t - r) + (Lj$)t(x,t - r)) dr)\\Mq,7) <

o j=1 (53)

< KQc1\\k\\Lp(Q y)Y 1-1/p(\\v\\H(qy) + \\$\\h(0 T)) < IIk\\Lp(Qy)c$Y1-1/p,

where c9 = kQc1(2Rc3 + \\$\\H(Q,T)). The estimates (51) - (53) ensure that

\\B-lA(k)\\Lp(0,y) < \\k\\Lp(Q,y)Yl-1/pc1Q, c1Q = (2c3(c6 + c8) + c9). (54)

Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming & Computer Software (Bulletin SUSU MMCS), 2018, vol. 11, no. 1, pp. 44-59

Choose Yi < 7o such that c107i-l/ = 1/2. In this case, for 7 < 71, the operator B-1F — B-lA(k) takes the ball BRR into itself. Demonstrate that it is also contractive for an appropriate 7. Let kl,k2 E BRR (7 < 7^. Denote by v1 ,v2 the corresponding solutions to the equation (42). We have

t 5 m

LoVi{x,t) = -fj E kj(r){LjVi)t- T) drdC-0 0 j=i

- E I kj (С) d^Lj (x, 0)uo(x) -ff E kj (r )(Lj 3>)t(x,£ - r) dr = G(v) г = 1, 2.

m t t 5 m

E/kj(С) d^Lj(x, 0)uo(x) - Ц j=i 0 00 j=i

Subtracting these equalities, we obtain that

t 5 m

(55)

L0vi-V2) = -Ц E(kj - kj)(r )(Lj vi)t(x, С - r) + k2(r)(Lj vi-Lj v2)(x,£-r) drd-

j- j j

0 0 j=i

m t t 5 m

(56)

-E I(k1 - k2m dL (x, 0)u0(x) -// J2(k} - kj)(T)(Lj $)t(x,£ - r) dr. j=i 0 00 j=i

As before in (48) (see also (45) and (47)), we have

\\vi-V2\\n(0Y) < Ьi-i/pCiC3R\\ki-k2\\Lp(0ri)+2c3\\ki-к2\\ьр(0Г() < \\ki-k2\\Lp(0r()4c3. (57)

Thus, we obtain one more additional summand as compared with the estimate (48). Next, consider the difference A(ki) - A(k2). We have

t m

Ai(ki) - Ai(k2) = *i(L-if E k)(r)(Ljvi)t(x,t - r) - k2(r)(Ljv2)t(x,t - r) dr) +

0 j=i

t m

+^i(L^iLt(vi - V2)) + %(v0(vi - V2)) + *i(L-if Z(kj - kj)(r)(LjФ)t(x,t - r) dr).

0 j=i

Repeating the arguments with use of the estimates (49), (57) (valid for the functions vi,v2), we conclude that

\\B-iA(ki) - B-iA(k2)\\Lp(0Y) < \\ki - k2\\Lp(0Y)Yi-i/pcii, cn = (cw + 4юс^3). (58)

Next, we can find y2 < y1 such th at Y12-1/vcii = 1/2. In this case, for 7 < j2, the equation (44) is uniquely solvable in the ball B\.

Proceed with the question of global (in time) solvability of (44). We argue as in the proof of Theorem 3. Demonstrate that there exists y3 < y2 such that the solvability of the system (44) on the segment [0,Iy3] (l = 1, 2,...,) implies the solvability of this system on [lY3,lY3 + r0], where r0 = min(73,T - Y)■ Assume that the system is solvable on [0,Iy3] (Y < T). Put

r( ) = { k(t), t E [lY3, lY3 + T0] yt) = { 0 t E [lY3, lY3 + r0] W I 0, t<lY3 ' W I k(t), t<lY3

The function k(t) is already known and we need to determine the function k(t). Using the notation S0(v) of Theorem 3, where y0 is replaced with y3, we rewrite (41) in the form

v + S0V = Si (k) + f3, (59)

where the right-hand side coincide with L° G(v) for t < Iy3 and, for t > Iy3, we have

t S m m t

S1(k) = -L-1// E kj(r)Lj(v + (x,£ - r) drd£ - E L-1 f kj(£) d£Lj(x, 0)uo(x), 0 0 j=1 j=1 0

t S m ^

fs = -L-1// E kj(r)Lj(v + $)c(x, £ - r) drd£-0 0 j=1

m t ^

-L-1 E I kj(£) d£Lj(x, 0)uo(x) + So(v). j=1 o

Note that the function f3 is calculated with the use of the values of the functions v, k on the segment [0, Iy2] and thus we can assume that it is a known function. We can see from

v kk

and the definition of the quantity y2 imply that

\\SQv\\H(Q,iy3+to) < c1Ys l/P\\k\\Lp(Q,y3)\v\h(Y,1^+to) < \\v\\H(lri, 113+to)/2. (6°)

Let v1 = (I + Sq)-1S1 (k). The estimate (60) yields \\(I + Sq)-1v\\h(q,y3+t0) < 2\\v\\h(q,ll3 +t0) for all v E H(0, h(3 + rQ). In this case Corollary 1, Lemma 3, and (47) - (49) imply that

\\v1\\h (0 ,lTi+T0) < 2ciyi l/P\\k\\Lp(ll3 Y +to)( \ \v \ \ H(0 Y3) + \mh(q ,l3)) + + 2c2\\k\\Lp(l13lY3+To) < 4cs\\k\\Lp(l13Iy3+to).

(61)

Hence, the function v = v1 + (I + SQ) 1f3 is estimated by the quantity

\\v\\H(Q,iy3+to) < 4c3\\k\\Lp(l13,ll3+To) + 2\\f3\\H(Q,lj3+To). (62)

Write out the representation for Aj ( k) for t > Iy3■ We have

Ai(k) = Ai(k) + f4i, Ai(k) = Vi(L-1Lotv1 + vo(v1))+

t m t-lj3

m

+Vi(L-1f E kj(r)Lj(v + $)tXx, t - r) dr) + ViL-1 f E kj(r)(Ljvx\(x, t - r) dr), 0 j=1 0 j=1

f4i = V i(L-1 Lot(I + So)-1f3) + Vi(vo((I + So )-1fs)) +

t m t m

+Vi(L-1f E kj(r)(LjQ)t(x, t - r) dr) + ViL-1 f E kj(r)(Ljv)t(x, t - r) dr) +

0 j=1 t-Y3l j=1

t-lj3 m ^

Vi(L- J E kj(r)(Lj(I + So)-1f3)t(x,t - r) dr). o j=1

The function f4i depends on the values of k on the segment (0,Iy3)- The remaining

kk

the support of v1 belongs to the segment [Iy2,Iy2 + rQ]. As before on the proof of the estimates (51) and (52), we derive that

Em=1(\\Vi(L - L0tv1\\Lp(lj3,lj3+To) + \ Vi(vQ(v1)) |Lp(l73,l73+To)) <

< (c6 + c8)Y3- \\ v1\h(iy3,iy3+to) < Y3 (c6 + c8)4c3\k\Lp(lY3,lY3+To).

Involving the arguments of the proof of (53), we infer

t m

-1

Tti Нф*(L- I E k!(t)((Ljv)t(x,t - T) + (LjФ)t(x,t - T)) dT)\\Lp{h3,h3+T0) < / л 0 j=i (64)

< K0CiYl~l/p\\k\\Lv(i13,17з+то)(\М\н(о ,7з) + \\ф\\н(о,t)) < Y¡~1/p\\k\\Lp(iTi,г13+т0)С9.

Moreover, we have

t—lj3 m

-1

YZi 11^(^0 1 / E^j (T)((Ljv1 )t(x,t — t) dT)\\Lp{iY3,lY3+T0) < , x

o j=i (65)

< KoCillol/p\\k\Lp(0,ri)4c3\k\Lp(iri,iri+T0) < k0CiyIo1/p2Rc3\\k\\Lp(lri,lTi+T0)-Thus, the estimate

\\BolA(k)\Lp(lYs,lY3+To) < 7i0l/P\\k\\Lp(lYslY3+To)((c6 + c8)4c3 + c9 + K0c12Rc3)

is valid. Hence, if we choose y3 so that

llo1/P (c + c8)4c3 + cg + Koci2Rc3) = 1/2,

then the operator Bo1A(k) is contractive. Therefore, if the system (44) is solvable on the segment [0,/y3] (/ = 1, 2,...,) then the system is solvable on [l^3,lY3 + t0], where t0 = min(Y3,T — ¡y3). The latter implies that the system (44) is solvable on [0,T], Show that the corresponding function v = v(k) (a solution to the problem (37), (38)) meets

L0

functional to (40), we obtain that

t m

(%(v))t + Vi(L01L0tv) + ^iLO1! E kj(t)(Ljv)t(x, t — t) dT) + %(v0) =

0 j=1

m t m

= — E kj№%(L~o1Lj(x, 0)u0(x)) — ^%(L~o1$ E kj(t)(Lj$)t(x,t — t) dT). j=1 0 j=1

Subtracting this equality from (43), we arrive at the equality (^(v))t — ijit = 0, or (in view of (13), (17)) ^¿(v) = i- e., the equalities (39) are fulfilled. The claim follows from Theorem 2.

The proof of the estimate (36) is in line with the proof of the existence. We just repeat the arguments paying attention to constants in the corresponding estimates.

□

Acknowledgement. The authors were supported by RFBR (Grant 18-01-00620) and by the Act 211 of the Government of the Russian Federation (contract 02.AOS.21.0011).

References

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Received January 30, 2018

УДК 517.956 DOI: 10.14529/mmpl80105

ОБРАТНЫЕ ЗАДАЧИ ДЛЯ МАТЕМАТИЧЕСКИХ МОДЕЛЕЙ КВАЗИСТАЦИОНАРНЫХ ЭЛЕКТРОМАГНИТНЫХ ВОЛН В АНИЗОТРОПНЫХ НЕМЕТАЛЛИЧЕСКИХ СРЕДАХ С ДИСПЕРСИЕЙ

С.Г. Пятков1'2, С. Н. Шергин1

1Югорский государственный университет, г. Ханты-Мансийск,

Российская Федерация

2

Российская Федерация

В работе рассматриваются обратные задачи эволюционного типа для математических моделей квазистационарных электромагнитных волн. В модели предполагается, что длина волны мала по сравнению с пространственными неоднородностями. Вводя электрический и магнитный потенциал получаем эллиптическое уравнение второго порядка по пространственным переменным, содержащее интегральные слагаемые типа свертки по времени. После дифференцирования по времени задача сводится к уравнению составного типа с интегральным слагаемым. Определению вместе с решением подлежат неизвестные коэффициенты в интегральном операторе. Дополнительно к краевым условиям задаются условия переопределения в виде заданного набора функционалов от решения, которые могут иметь произвольный вид (интегралы от решения с весом, значения решения в отдельных точках и пр.). В качестве основных пространств рассматриваются пространства С.Л. Соболева. Доказываются теоремы о существовании и единственности решения поставленной задачи в целом по времени, приводится оценка устойчивости.

Ключевые слова: уравнения соболевского типа; эллиптическое уравнение; уравнения с памятью; обратная задача; краевая задача.

Литература

1. Свешников, А.Г. Линейные и нелинейные уравнения соболевского типа / А.Г. Свешников, А.Б. Алынин, М.О. Корпусов, Ю.Д. Плетнер. - М.: Физматлитература, 2007.

2. Рабов, С.А. Линейные задачи теории нестационарных внутренних волн / С.А. Рабов, А.Г. Свешников. - М.: Наука, 1990.

3. Lorenzi, A. Direct and Inverse Problems in the Theory of Materials with Memory / A. Lorenzi, I. Paparone // Rendiconti del Seminario matemático délia Universita di Padova. - 1992. -V. 87. - P. 105-138.

4. Jarmo, J. Inverse Problems for Identification of Memory Kernels in Viscoelasticity / J. Janno, L. Von Wolfersdorf // Mathematical Methods in the Applied Sciences. - 1997. - V. 20. -P. 291-314.

5. Durdiev, D.K. Inverse Problem of Determining the One-Dimensional Kernel of the Viscoelasticity Equation in a Bounded Domain / D.K. Durdiev, Zh.Sh. Safarov // Mathematical Notes. - 2015. - V. 97, № 6. - P. 867-877.

6. Colombo, F. An Inverse Problem for a Phase-Field Model in Sobolev Spaces. Nonlinear Elliptic and Parabolic Problems / F. Colombo, D. Guidetti // Progress in Nonlinear Differential Equations and Their Applications. - V. 64. - Basel: Birkhäuser Verlag, 2005. -P. 189-210.

7. Guidetti, D. A Mixed Type Identification Problem Related to a Phase-Field Model with Memory / D. Guidetti, A. Lorenzi // Osaka Journal of Mathematics. - 2007. - V. 44. -P. 579-613.

8. Colombo, F. A Global in Time Existence and Uniqueness Result for a Semilinear Integrodifferential Parabolic Inverse Problem in Sobolev Spaces / F. Colombo, D. Guidetti // Mathematical Models and Methods in Applied Sciences. - 2007. - V. 17, № 4. - P. 537-565.

9. Коломбо, Ф. О некоторых методах решения интегрально-дифференциальных обратных задач параболического типа / Ф. Коломбо // Вестник ЮУрГУ. Серия: Математическое моделирование и программирование. - 2015. - Т. 8, № 3. - С. 95-115.

10. Favini, A. Identication Problems for Singular Integro-Differential Equations of Parabolic Type / A. Favini, A. Lorenzi // Nonlinear Analysis. - 2004. - V. 56, № 6. - P. 879-904.

11. Lorenzi, A. Inverse and Direct Problems for Nonautonomous Degenerate Integro-Differential Equations of Parabolic Type with Dirichlet Boundary Conditions / A. Lorenzi, H. Tanabe // Differential Equations: Inverse and Direct Problems. Lecture Notes in Pure and Applied Mathematics. - Boca Raton, London, N.Y.: Chapman and Hall CRC Taylor and Francis Group, 2006. - V. 251. - P. 197-244.

12. Abaseeva, N. Identification Problems for Nonclassical Integro-Differential Parabolic Equations / N. Abaseeva, A. Lorenzi // Journal of Inverse and Ill-Posed Problems. - 2005. -V. 13, № 6. - P. 513-535.

13. Асанов, А. Обратная задача для операторного интегро-дифференциального псевдопараболического уравнения / А. Асанов, Э.Р. Атаманов // Сибирский математический журнал. - 1995. - Т. 36, № 4. - С. 752-762.

14. Avdonin, S.A. Inverse Problems for the Heat Equation with Memory / S.A. Avdonin, S.A. Ivanov, J. Wang. - 2017. - 10 p. - URL: https://arxiv.org/abs/1612.02129 (дата обращения: 9 февраля 2018 г.)

15. Pandolfi, L. Identification of the Relaxation Kernel in Diffusion Processes and Viscoelasticity with Memory via Deconvolution / L. Pandolfi. - 2016. - 15 p. - URL: https://arxiv.org/abs/1603.04321 (дата обращения: 9 февраля 2018 г.)

16. Денисов, A.M. Обратная задача для квазилинейного интегро-дифференциального уравнения / A.M. Денисов // Дифференциальные уравнения. - 2001. - Т. 37, № 10. -С. 1350-1356.

17. Triebel, Н. Interpolation Theory. Function Spaces. Differential Operators / H. Triebel. -Berlin: VEB Deutscher Verlag der Wissenschaften, 1978.

18. Ладыженская, O.A. Линейные и квазилинейные уравнения эллиптического типа / O.A. Ладыженская, H.H. Уральцева. - М.: Наука, 1973.

19. Гилбарг, Д. Эллиптические дифференциальные уравнения с частными производными второго порядка / Д. Гилбарг, Н. Трудингер. - М.: Наука, 1989.

20. Maugeri, A. Elliptic and Parabolic Equations with Discontinuous Coefficients / A. Maugeri, D.K. Palagachev, L.G. Softova. - Berlin: Wiley-VCH Verlag, 2000.

Сергей Григорьевич Пятков, доктор физико-математических наук, заведующий кафедрой «Высшая математика:», Югорский государственный университет (г. Ханты-Мансийск, Российская Федерация); научно-исследовательская лаборатория «Неклассические уравнения математической физики», Южно-Уральский государ-ствбнныи университет (г. Челябинск, Российская Федерация), [email protected].

Сергей Николаевич Шергин, аспирант, кафедра «Высшая математика», Югорский государственный университет (г. Ханты-Мансийск, Российская Федерация), [email protected].

Поступила в редакцию 30 января 2018 г.